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arXiv:2506.19965 · Communications Physics

Unlocking multidimensional integration with quantum adaptive importance sampling

Konstantinos Pyretzidis, Jorge J. Martínez de Lejarza, Germán Rodrigo A

Published May 22, 2026• 0 citations
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Abstract

Multidimensional numerical integration is a central ingredient of theoretical predictions in high-energy physics, where multiloop Feynman diagrams and phase-space integrals are computationally demanding due to divergences and complex mathematical structures. Established Adaptive Importance Sampling methods for numerical integration, such as VEGAS, iteratively refine a grid in a separable way, dimension by dimension. This keeps the algorithm scalable but reduces performance when strong inter-variable correlations are present. In this work, we introduce a hybrid quantum-classical algorithm that performs Quantum Adaptive Importance Sampling (QAIS) for multidimensional Monte Carlo integration. Our approach uses a Parametrized Quantum Circuit to encode a non-separable Probability Density Function on a multidimensional grid and allocate samples efficiently in the integration domain. We apply the method to a sharply peaked loop Feynman integral and to multi-modal benchmark integrals. Our results show that QAIS provides an efficient route for high-precision evaluation of multidimensional integrals. Theoretical calculations in particle physics rely on multidimensional integrals that become difficult to evaluate in the presence of correlations. Here, the authors show that a hybrid quantum-classical algorithm implementing Quantum Adaptive Importance Sampling can learn correlated structures efficiently and deliver precise results for nontrivial integrals.

DOI
10.1038/s42005-026-02684-7

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Multidimensional numerical integration is a central ingredient of theoretical predictions in high-energy physics, where multiloop Feynman diagrams and phase-space integrals are computationally demanding due to divergences and complex mathematical structures. Established Adaptive Importance Sampling methods for numerical integration, such as VEGAS, iteratively refine a grid in a separable way, dimension by dimension. This keeps the algorithm scalable but reduces performance when strong inter-variable correlations are present. In this work, we introduce a hybrid quantum-classical algorithm that performs Quantum Adaptive Importance Sampling (QAIS) for multidimensional Monte Carlo integration. Our approach uses a Parametrized Quantum Circuit to encode a non-separable Probability Density Function on a multidimensional grid and allocate samples efficiently in the integration domain. We apply the method to a sharply peaked loop Feynman integral and to multi-modal benchmark integrals. Our results show that QAIS provides an efficient route for high-precision evaluation of multidimensional integrals. Theoretical calculations in particle physics rely on multidimensional integrals that become difficult to evaluate in the presence of correlations. Here, the authors show that a hybrid quantum-classical algorithm implementing Quantum Adaptive Importance Sampling can learn correlated structures efficiently and deliver precise results for nontrivial integrals.
This work presents Quantum Adaptive Importance Sampling (QAIS), a hybrid quantum-classical algorithmic workflow that performs Monte Carlo numerical integration of multivariate functions. This workflow extends standard Importance Sampling to a quantum computational system. Its primary purpose is to produce an unbiased estimate of integrals, with a limited number of measurements. By using the exponentially sized Hilbert space of a Parameterized Quantum Circuit (PQC), we manipulate the Probability Density Function (PDF) over a grid in its entirety. With adequate expressivity and entanglement, we effectively capture correlations among different integrand variables, thus bypassing the separable PDF assumption, as implemented in VEGAS. By optimizing the PQC, we adapt and ultimately load a PDF that approximates the integrand's behavior. Direct sampling from it, allocates samples to the important regions, allowing a highly accurate integral estimation. As an application, we study a benchmark two-dimensional integral, and compare the precision of the best grid of VEGAS against that of the quantum-generated proposal PDF, as a function of the number of samples.
Integration and sampling in high dimensions are among central challenges in modern science and technology, underlying applications ranging from quantum many-body physics to Bayesian inference and artificial intelligence. Although conventional Monte Carlo methods are formally scalable, their efficiency deteriorates rapidly in the presence of strong correlations or sharp features in high-dimensional configuration space. Here we introduce a new framework, termed the Schr"odinger Generator, for integration and sampling based on the explicit optimization of coordinate transformations. The method decomposes the total Jacobian into two complementary components, including an adaptive map that minimizes estimator variance by learning the marginal structure in each dimension, and a normalizing-flow-based transformation that captures non-factorizable correlations in the target distribution. A final resampling step guarantees unbiased sampling even when the learned transformation is imperfect. We demonstrate stable and scalable performance for nuclear quantum many-body states in dimensions exceeding 600. Short-range correlations among nucleons in finite nucleus are faithfully reproduced. The framework offers a physically transparent approach to high-dimensional stochastic integration and sampling, opening new possibilities for simulations of complex quantum systems.
This work investigates in detail the performance and advantages of a new quantum Monte Carlo integrator, dubbed quantum Fourier iterative amplitude estimation (QFIAE), to numerically evaluate for the first time loop Feynman integrals in a near-term quantum computer and a quantum simulator. In order to achieve a quadratic speedup, QFIAE introduces a quantum neural network (QNN) that efficiently decomposes the multidimensional integrand into its Fourier series. For a one-loop tadpole Feynman diagram, we have successfully implemented the quantum algorithm on a real quantum computer and obtained a reasonable agreement with the analytical values. One-loop Feynman diagrams with more external legs have been analyzed in a quantum simulator. These results thoroughly illustrate how our quantum algorithm effectively estimates loop Feynman integrals and the method employed could also find applications in other fields such as finance, artificial intelligence, or other physical sciences. Published by the American Physical Society 2024
Monte Carlo integration using quantum computers has been widely investigated, including applications to concrete problems. It is known that quantum algorithms based on quantum amplitude estimation (QAE) can compute an integral with a smaller number of iterative calls of the quantum circuit which calculates the integrand, than classical methods call the integrand subroutine. However, the issues about the iterative operations in the integrand circuit have not been discussed so much. That is, in the high-dimensional integration, many random numbers are used for calculation of the integrand and in some cases similar calculations are repeated to obtain one sample value of the integrand. In this paper, we point out that we can reduce the number of such repeated operations by a combination of the nested QAE and the use of pseudorandom numbers (PRNs), if the integrand has the separable form with respect to contributions from distinct random numbers. The use of PRNs, which the authors originally proposed in the context of the quantum algorithm for Monte Carlo, is the key factor also in this paper, since it enables parallel computation of the separable terms in the integrand. Furthermore, we pick up one use case of this method in finance, the credit portfolio risk measurement, and estimate to what extent the complexity is reduced.
High-quality random samples of quantum states are needed for a variety of tasks in quantum information and quantum computation. Searching the high-dimensional quantum state space for a global maximum of an objective function with many local maxima or evaluating an integral over a region in the quantum state space are but two exemplary applications of many. These tasks can only be performed reliably and efficiently with Monte Carlo methods, which involve good samplings of the parameter space in accordance with the relevant target distribution. We show how the Markov-chain Monte Carlo method known as Hamiltonian Monte Carlo, or hybrid Monte Carlo, can be adapted to this context. It is applicable when an efficient parameterization of the state space is available. The resulting random walk is entirely inside the physical parameter space, and the Hamiltonian dynamics enable us to take big steps, thereby avoiding strong correlations between successive sample points while enjoying a high acceptance rate. We use examples of single and double qubit measurements for illustration.